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Mirrors > Home > MPE Home > Th. List > rspsn | Structured version Visualization version GIF version |
Description: Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
rspsn.b | ⊢ 𝐵 = (Base‘𝑅) |
rspsn.k | ⊢ 𝐾 = (RSpan‘𝑅) |
rspsn.d | ⊢ ∥ = (∥r‘𝑅) |
Ref | Expression |
---|---|
rspsn | ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2831 | . . . . 5 ⊢ (𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ (𝑎(.r‘𝑅)𝐺) = 𝑥) | |
2 | 1 | a1i 11 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
3 | 2 | rexbidv 3261 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺) ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
4 | rlmlmod 19565 | . . . 4 ⊢ (𝑅 ∈ Ring → (ringLMod‘𝑅) ∈ LMod) | |
5 | rlmsca2 19561 | . . . . 5 ⊢ ( I ‘𝑅) = (Scalar‘(ringLMod‘𝑅)) | |
6 | baseid 16281 | . . . . . 6 ⊢ Base = Slot (Base‘ndx) | |
7 | rspsn.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 6, 7 | strfvi 16275 | . . . . 5 ⊢ 𝐵 = (Base‘( I ‘𝑅)) |
9 | rlmbas 19555 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘(ringLMod‘𝑅)) | |
10 | 7, 9 | eqtri 2848 | . . . . 5 ⊢ 𝐵 = (Base‘(ringLMod‘𝑅)) |
11 | rlmvsca 19562 | . . . . 5 ⊢ (.r‘𝑅) = ( ·𝑠 ‘(ringLMod‘𝑅)) | |
12 | rspsn.k | . . . . . 6 ⊢ 𝐾 = (RSpan‘𝑅) | |
13 | rspval 19553 | . . . . . 6 ⊢ (RSpan‘𝑅) = (LSpan‘(ringLMod‘𝑅)) | |
14 | 12, 13 | eqtri 2848 | . . . . 5 ⊢ 𝐾 = (LSpan‘(ringLMod‘𝑅)) |
15 | 5, 8, 10, 11, 14 | lspsnel 19361 | . . . 4 ⊢ (((ringLMod‘𝑅) ∈ LMod ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ ∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺))) |
16 | 4, 15 | sylan 577 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ ∃𝑎 ∈ 𝐵 𝑥 = (𝑎(.r‘𝑅)𝐺))) |
17 | rspsn.d | . . . . 5 ⊢ ∥ = (∥r‘𝑅) | |
18 | eqid 2824 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
19 | 7, 17, 18 | dvdsr2 19000 | . . . 4 ⊢ (𝐺 ∈ 𝐵 → (𝐺 ∥ 𝑥 ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
20 | 19 | adantl 475 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐺 ∥ 𝑥 ↔ ∃𝑎 ∈ 𝐵 (𝑎(.r‘𝑅)𝐺) = 𝑥)) |
21 | 3, 16, 20 | 3bitr4d 303 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝑥 ∈ (𝐾‘{𝐺}) ↔ 𝐺 ∥ 𝑥)) |
22 | 21 | abbi2dv 2946 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝐺 ∈ 𝐵) → (𝐾‘{𝐺}) = {𝑥 ∣ 𝐺 ∥ 𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {cab 2810 ∃wrex 3117 {csn 4396 class class class wbr 4872 I cid 5248 ‘cfv 6122 (class class class)co 6904 ndxcnx 16218 Basecbs 16221 .rcmulr 16305 Ringcrg 18900 ∥rcdsr 18991 LModclmod 19218 LSpanclspn 19329 ringLModcrglmod 19529 RSpancrsp 19531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-er 8008 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-sca 16320 df-vsca 16321 df-ip 16322 df-0g 16454 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-grp 17778 df-minusg 17779 df-sbg 17780 df-subg 17941 df-mgp 18843 df-ur 18855 df-ring 18902 df-dvdsr 18994 df-subrg 19133 df-lmod 19220 df-lss 19288 df-lsp 19330 df-sra 19532 df-rgmod 19533 df-rsp 19535 |
This theorem is referenced by: lidldvgen 19615 zndvds 20256 |
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